Classifying Hopf algebras of a given dimension
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Classifying all Hopf algebras of a given finite dimension over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a challenging problem which remains open even for many small dimensions, not least because few general approaches to the problem are known. Some useful techniques include counting the dimensions of spaces related to the coradical filtration in D. Fukuda (Glasg. 2008), N. Andruskiewitsch and S. Natale (2001), M. Beattie and S. Dăscălescu (2004), studying sub- and quotient Hopf algebras in G.A. Garcia (2005), G.A. Garcia and C. Vay (2010), especially those sub-Hopf algebras generated by a simple subcoalgebra in S. Natale (2002), working with the antipode in S-H. Ng (2002), (2004), (2005), (2008), and studying Hopf algebras in Yetter-Drinfeld categories to help to classify Radford biproducts in Y-l. Cheng and S-H. Ng (2011). In this paper, we add to the classification tools in M. Beattie and G.A. Garcia (to appear) and apply our results to Hopf algebras of dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r p q"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mi>p</mml:mi> <mml:mi>q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">rpq</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="8 p"> <mml:semantics> <mml:mrow> <mml:mn>8</mml:mn> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">8p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p comma q comma r"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p,q,r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are distinct primes. At the end of this paper we summarize in a table the status of the classification for dimensions up to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="100"> <mml:semantics> <mml:mn>100</mml:mn> <mml:annotation encoding="application/x-tex">100</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to date.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it